From Bilbo's departure in The Hobbit, it follows that going on a trip without a hat was unusual. Hats were commonplace in the Shire: for example, when Pippin and Merry returned to the Shire as small giants, they were asked about their hat sizes.

However, I don't remember the novel ever referring to the hats that Hobbits should wear during their adventure.

"The Shirriffs was the name that the Hobbits gave to their police, or the nearest equivalent that they possessed. They had, of course, no uniforms (such things being quite unknown), only a feather in their caps; and they were in practice rather haywards than policemen, more concerned with the strayings of beasts than of people. There were in all the Shire only twelve of them, three in each Farthing, for Inside Work. A rather larger body, varying at need, was employed to 'beat the bounds', and to see that Outsiders of any kind, great or small, did not make themselves a nuisance."
– ValorumAug 14 '18 at 21:42

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Just so you know, this isn't the final form I hope the answer to take, when I get some more time to plan I hope to edit it and make it neater.
– EdlothiadAug 16 '18 at 8:31

@Edlothiad OK. Am I suposed to "unaccept" your answer until you improve it, or is fine to leave it like this? :-)
– GinasiusAug 16 '18 at 13:15

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Nah you can leave it, I was just letting you know. The content is staying the same, I just want to restructure because I feel it is messy.
– EdlothiadAug 16 '18 at 13:20

2 Answers
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There is no mention of Hobbits wearing hats in the Lord of the Rings

I have searched my copies for the words "hat" and "hats", and never are the Hobbits mentioned to be wearing hats during the events of the Lord of the Rings.1 As such, it seems unclear whether or not they were wearing or weren't wearing hats.

NB: This does not mean that Hobbits, in their daily lives did not wear hats, such as the Shirrifs pointed out by Valorum.

There are two characters in the Lord of the Rings that wear hats. Namely, Gandalf and Tom Bombadillo.

There was another burst of song, and then suddenly, hopping and dancing along the path, there appeared above the reeds an old battered hat with a tall crown and a long blue feather stuck in the band.Book 1, Chapter VI: The Old Forest

Arrows fell among them. One struck Frodo and sprang back. Another pierced Gandalf ’s hat and stuck there like a black feather.Book 2, Chapter V: The Bridge Of Khazad-Dûm

Gandalf's hat is mentioned a few more times in The Two Towers, with no mention of hats in the Return of the King, before the chapter Many Partings.

The only mention of Hobbits and hats are the following three:

'Don’t let your heads get too big for your hats! But if you don’t finish growing up soon, you are going to find hats and clothes expensive.’Book 6, Chapter VI: Many Partings

He opened an eye and tried gallantly to smile. ‘Who’s this young giant with the loud voice?’ he whispered. ‘Not little Pippin! What’s your size in hats now?’Book 6, Chapter IX: The Grey Havens

Sam's felt "hat"

In Three is Company, before Pippin, Frodo and Sam set out for Frodo's new home, Sam appears with a "tall shapeless felt bag" on his head, which he calls a "hat". This is quite an interesting point, given the difference in status between the other two adventurers, Pippin and Frodo, and Sam. Sam is a lower class, more simple Hobbit. Gardener for Frodo, Sam may have chucked on the felt bag to feel more akin with his fellow travellers, who would have normally been seen with a cap of some kind.

Similarly, in the History of the Lord of the Rings, apart from the earliest works, the story talks of no mention of hats being worn by the Hobbits in the Fellowship. Although there is a very early version of the tale, where Frodo wears the Ring to escape from Maggot, and in the process temporarily steals Farmer Maggots hat.

Bar this, there is no mention of the Hobbits adventuring with a hat.

Some pictorial evidence from renowned Tolkien artists Alan Lee and John Howe, which may help shape viewpoints, show that the Hobbits likely were not wearing hats during their adventures:

1 Did you search your copies for homburg, fedora, bycocket or bicoket, bicorne or tricorne, chapeau, bonnet, beret, biretta, boater (I just know there was a straw boater in Bored of the Rings), bowler, capotain, caubeen, cloche, deerstalker, fez, gatsby, hardee, kofia, porkpie, shako, slouch, taqiyah, toque, trilby, vueltiao, or any other of the numerous kinds of hats that do not bear 'hat' in their name? ;)Comment by @Lexible

Based on this comment by @Lexible, stating my search terms were far too limited, I dove deeper, my results are below:

I think you have your assumptions backwards. For someone writing in the 1940s - particularly one born in the 1890s - the wearing of hats out-of-doors would be an assumed thing. The mentionable thing would be if they didn't. Therefore - by your reasoning that Tolkien wouldn't have missed details - we can assume that if he didn't mention them being bare-headed, we can conclude that they did indeed have something on their head.
– R.M.Aug 15 '18 at 0:56

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@R.M.: To second that. Thorin Oakenshield's hat in the Hobbit is only mentioned when it's useful to the plot. Several other hats appear in the surrounding paragraphs and are promptly forgotten again.
– JoshuaAug 15 '18 at 4:12

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Not to mention that casually asking someone's hat size suggests that you expect him to know it (most likely because he frequently wears hats).
– chrylisAug 15 '18 at 6:37

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What about Samwise Gamgee? In the chapter Three is Company, it is stated that he "put on his head a tall shapeless felt bag, which he called a hat".
– J WAug 15 '18 at 6:52

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@JW I chose these artists specifically because of their correspondence with Christopher Tolkien, and their repeated contracts to work on Tolkien's works. It is merely representative. I believe my stance is poorly portrayed in this answer, as I'm not strongly gunning for one side or the other. Merely that no hats were mentioned, so it's really up to you, as RM shows.
– EdlothiadAug 15 '18 at 8:00

As Edlothiad had shown, there is no positive evidence that they wore hats.

I think that the fact that hats are not mentioned where it would be most logical to mention them makes it very probable that they were not there.

For example during the fight in Chamber of Mazarbul Sam recieves a light head wound from an orc blow. Would he had a hat on, somone (himself and narrator included) would remark on the damage to it. And when he splashed into the Anduin at Tol Brandir, it would float away.

Also they never use hats to wave for attention, nor are they lifted in greeting.

In conclusion, it seems plain to me that hats are usually worn by Hobbits while working outdoors, taking trips and walks in the Shire or participating on outdoor social events, but Frodo and his companions left them intentionally behind, knowing that these items would be a hindrance on the journey, and would soon be lost or destroyed. They probably used hooded cloaks instead.

Absence of a mention of a thing isn't proof of its absence. I'm struggling to remember any time that you see an elf pooping but that doesn't mean they don't poop
– ValorumAug 21 '18 at 18:47

Of course not. If it would be, my answer would be superfluous, Edlothiad having settled the case. What I am trying to say that if Tolkien had imagined the hobbits to go journeing in hats (but did not told us because it was self-evident to him) he would likely still mention them when there is meaningful action related to them. As another example, Rohhirim probably do not wear hats. When Merry and Pippin greet Theoden, the cultural differences are emphasized in the narrative. Would they have hats, they would be described to lift and wave them in formal greeting
– b.LorenzAug 21 '18 at 19:00

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In 'Fog on the Barrow-Downs' it is explicitly mentioned "they halted and got out cloaks and hoods". Just prior to this the narrator mentions that "their hair hung lank and dripping on their foreheads". If they were wearing hats it is unlikely their hair would be hanging loose across their foreheads. Merry is described as being "cloaked", with a scarf, when he approaches Farmer Maggot and the others. In the Midgewater marshes the midges got into the Hobbits' hair (unlikely with hats on). After Lorien they had cloaks, but I can't find mention that they had hoods. Later some of them wore helms.
– David RobertsAug 22 '18 at 1:42

As we saw from scifi.stackexchange.com/a/132077/36526, Tolkien doesn't seem to have cared a lot about clothing in general. The absence of hats in the written text does not necessarily mean a thing, aside from Tolkien neglecting to discuss it.
– KevinOct 11 '18 at 6:02